Theorem sbcom2 1863

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2	⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Distinct variable groups:   𝑥,𝑧   𝑥,𝑤   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem sbcom2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2v2 1862	. . . 4 ⊢ ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
2	1	sbbii 1648	. . 3 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑)
3		sbcom2v2 1862	. . 3 ⊢ ([𝑤 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
4	2, 3	bitri 173	. 2 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑)
5		ax-17 1419	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑣[𝑦 / 𝑥]𝜑)
6	5	sbco2v 1821	. 2 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
7		ax-17 1419	. . . 4 ⊢ (𝜑 → ∀𝑣𝜑)
8	7	sbco2v 1821	. . 3 ⊢ ([𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑)
9	8	sbbii 1648	. 2 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑣][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
10	4, 6, 9	3bitr3i 199	1 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Syntax hints:   ↔ wb 98  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  2sb5rf  1865  2sb6rf  1866  sbco4lem  1882  sbco4  1883  sbmo  1959  cnvopab  4726